The Impossibility of Approximate Agreement on a Larger Class of Graphs

Author Shihao Liu

Thumbnail PDF


  • Filesize: 0.69 MB
  • 20 pages

Document Identifiers

Author Details

Shihao Liu
  • Department of Computer Science, University of Toronto, Canada


I want to thank my advisor Faith Ellen for her advice and many helpful discussions. I would also like to thank the anonymous reviewers for their helpful comments.

Cite AsGet BibTex

Shihao Liu. The Impossibility of Approximate Agreement on a Larger Class of Graphs. In 26th International Conference on Principles of Distributed Systems (OPODIS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 253, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Approximate agreement is a variant of consensus in which processes receive input values from a domain and must output values in that domain that are sufficiently close to one another. We study the problem when the input domain is the vertex set of a connected graph. In asynchronous systems where processes communicate using shared registers, there are wait-free approximate agreement algorithms when the graph is a path or a tree, but not when the graph is a cycle of length at least 4. For many graphs, it is unknown whether a wait-free solution for approximate agreement exists. We introduce a set of impossibility conditions and prove that approximate agreement on graphs satisfying these conditions cannot be solved in a wait-free manner. In particular, the graphs of all triangulated d-dimensional spheres that are not cliques, satisfy these conditions. The vertices and edges of an octahedron is an example of such a graph. We also present a family of reductions from approximate agreement on one graph to another graph. This allows us to extend known impossibility results to even more graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Computability
  • Approximate agreement on graph
  • wait-free solvability
  • triangulated sphere


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Manuel Alcántara, Armando Castañeda, David Flores-Peñaloza, and Sergio Rajsbaum. The topology of look-compute-move robot wait-free algorithms with hard termination. Distrib. Comput., 32(3):235-255, 2019. Google Scholar
  2. Dan Alistarh, James Aspnes, Faith Ellen, Rati Gelashvili, and Leqi Zhu. Why extension-based proofs fail. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing, (STOC), pages 986-996, 2019. Google Scholar
  3. Dan Alistarh, Faith Ellen, and Joel Rybicki. Wait-free approximate agreement on graphs. In Proceeding of the 28th International Colloquium on Structural Information and Communication Complexity, (SIROCCO), pages 87-105, 2021. Google Scholar
  4. Dan Alistarh, Faith Ellen, and Joel Rybicki. Wait-free approximate agreement on graphs, 2021. URL:
  5. Hagit Attiya and Armando Castañeda. A non-topological proof for the impossibility of k-set agreement. In Proceedings of the 13th International Symposium on Stabilization, Safety, and Security of Distributed Systems, (SSS), pages 108-119, 2011. Google Scholar
  6. Hagit Attiya and Faith Ellen. Impossibility results for distributed computing. Morgan et Claypool Publishers, 2014. Google Scholar
  7. Hagit Attiya and Faith Ellen. The step complexity of multidimensional approximate agreement. In Proceedings of the 26th International Conference on Principles of Distributed Systems, (OPODIS), pages 25:1-25:12, 2022. Google Scholar
  8. Hagit Attiya, Nancy Lynch, and Nir Shavit. Are wait-free algorithms fast? J. ACM, 41(4):725-763, 1994. Google Scholar
  9. Elizabeth Borowsky and Eli Gafni. Generalized flp impossibility result for t-resilient asynchronous computations. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, (STOC), pages 91-100. Association for Computing Machinery, 1993. Google Scholar
  10. Armando Castañeda, Sergio Rajsbaum, and Matthieu Roy. Convergence and covering on graphs for wait-free robots. Journal of the Brazilian Computer Society, 24(1):1-15, 2018. Google Scholar
  11. Danny Dolev, Nancy Lynch, Shlomit Pinter, Eugene Stark, and William Weihl. Reaching approximate agreement in the presence of faults. Journal of the ACM, 33(3):499-516, 1986. Google Scholar
  12. Michael J. Fischer, Nancy A. Lynch, and Michael S. Paterson. Impossibility of distributed consensus with one faulty process. J. ACM, 32(2):374-382, 1985. Google Scholar
  13. Maurice Herlihy, D. N. Kozlov, and Sergio Rajsbaum. Distributed computing through combinatorial topology. Morgan Kaufmann, 2014. Google Scholar
  14. Maurice Herlihy and Nir Shavit. The topological structure of asynchronous computability. J. ACM, 46(6):858-923, 1999. Google Scholar
  15. Gunnar Hoest and Nir Shavit. Toward a topological characterization of asynchronous complexity. SIAM Journal on Computing, 36(2):457-497, 2006. Google Scholar
  16. Jérémy Ledent. Brief announcement: variants of approximate agreement on graphs and simplicial complexes. In Proceedings of the 40th Annual ACM Symposium on Principles of Distributed Computing, (PODC), pages 427-430, 2021. Google Scholar
  17. Hammurabi Mendes, Maurice Herlihy, Nitin Vaidya, and Vijay K Garg. Multidimensional agreement in byzantine systems. Distributed Computing, 28(6):423-441, 2015. Google Scholar
  18. Thomas Nowak and Joel Rybicki. Byzantine Approximate Agreement on Graphs. In Proceedings of the 33rd International Symposium on Distributed Computing, (DISC), pages 29:1-29:17, 2019. Google Scholar
  19. M. Pease, R. Shostak, and L. Lamport. Reaching agreement in the presence of faults. J. ACM, 27(2):228-234, 1980. Google Scholar
  20. Michael Saks and Fotios Zaharoglou. Wait-free k-set agreement is impossible: The topology of public knowledge. SIAM Journal on Computing, 29(5):1449-35, 2000. Google Scholar
  21. E. Schenk. Faster approximate agreement with multi-writer registers. In Proceedings of the 36th IEEE Annual Symposium on Foundations of Computer Science, (FOCS), pages 714-723, 1995. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail